They do not seek atheism, they become atheists after the fact
Usually they do not like dogmas which tells them what is allowed and what's not.
What you are describing here is faith-based doctrine, in other words: theism (*).
When people are in conflict with a doctrine (no matter its origin) it is because their own values and/or their own reasoning do not arrive at the same conclusions as the doctrine. They have used their own experience, their own judgement, their own feelings, and their own ability to empathise with other people... and they have — through the use of all of those tools — arrived at something that clashes with the doctrine.
Normally, when someone is in conflict with a doctrine, they will seek out the discourse that led to the doctrine. If — for example — they do not like a certain law, they will debate the matter; they will get in touch with legislators; they will seek dialogue with their peers... all in the hope that they will either affect change, and/or to understand the arguments and reasoning that led to the doctrine, and thereby be able to accept it.
When doctrines are sourced in human values and human reasoning, people will — through discourse with their peers — often find acceptance or at least agree to disagree; they will tolerate the present doctrine. And even if the doctrine does clash with their own values, they can still operate within it without the doctrine grating too much on their own sensibilities. Being heard by your peers and accepting "defeat" is all right, if the discourse has been open, fair and equal.
Faith-based doctrine however very rarely offers open, fair and equal discourse. Faith-based doctrine often do not even allow for a discussion in the first place; in some places it is even illegal to engage in religious discourse unless it is to advocate for the doctrine in question. And when faith-based doctrine does offer a discourse you are not debating against your peers but instead you face arguments like "This is the divine will" / "This is the higher truth" / "This is what the authorities say". It becomes even worse if the ones in favour of the doctrine do not provide rational reasoning behind the arguments but instead say "You just have to take it on faith". You are being asked to reject everything you have reasoned and argued about the matter, and instead "just have faith" that what you believe and feel is all wrong; you have to just lie down flat and accept that the other side is right.
At this point there are those say "I surrender to the religious doctrine". But there are also those that cannot reconcile with this and therefore say "No, I will not accept that". The latter have then declared that they are not accepting the dogma; they are indifferent to/rejecting the faith-based doctrine. This — automatically — makes them atheists, at least in regards to that particular doctrine ( ** ).
Do note that theism and atheism are descriptive terms, not prescriptive dittos. You do not reject (a-) faith-based (-the-) doctrine (-ism) because you are an atheist; you are an atheist because you reject faith-based doctrine. And conversely: you do not accept dogma because you are a theist, you are a theist because you accept dogma.
In other words: people do not turn to atheism to escape religious dogma. Instead they become atheists once they have escaped the religious dogma.
In this process however, they may turn to atheists — in the meaning: approach atheists — in order to seek out arguments for and support in their struggle. Breaking up with a faith is usually traumatic and hard. In that process it helps to talk to other people that have gone through the same. So they are not turning to atheism, because atheism is not a doctrine, it is the rejection of doctrine. But they can turn to other people that reject the doctrine — atheists — and ask for advice.
So there is your answer: people that want to escape religious dogma do not turn to atheism, they turn to atheists. And they do this in order to seek support, arguments, and comfort during a time that — almost always — is very difficult and upsetting. It is easier to face such a process when in contact with people that have experienced the same situation and can empathise with it.
( * ) And before any commenters draw breath to say "No, theism is to believe in a god and atheism is to not believe in a god!", you must define what it means to "believe in a god".
( ** ) This is the source of the half joking/fully serious claim that "Everyone — including the faithful — knows what it is like to be an atheist. Us atheists just take it one god/faith further", because everyone is rejecting at least one faith-based doctrine.